# Solve for x. 7^x= 6^{x+7}

## Question:

Solve for x.

{eq}7^x= 6^{x+7} {/eq}

## Solving an Equation

We have to find a value of x for which both sides of the equation are equal. This can be done by using the natural log of both sides of the equation as this will allow us to move the x down.

The value of x can be found as follows by taking the natural log of everything.

\begin{align} 7^x&= 6^{x+7}\\ \Rightarrow \ln (7^x)&=\ln (6^{x+7})\\ x\ln 7&=(x+7)\ln 6&&&&\left [ \because \ln (a^b)=b\ln a \right ]\\ \frac{\ln 7}{\ln 6}&=\frac{x+7}{x}\\ 1.086&=1+\frac{7}{x}\\ \therefore x&\approx 81.40 \end{align}